Before beginning polynomial calculations, first master multiplication and division of monomials. We will also gradually use the laws of exponents.
Multiplication of Monomials
Multiply the coefficients and the variables separately. Combine the same variables into powers.
Example: 3x × 4y ＝ 12xy、2a × 5a ＝ 10a²、(−3x)×(4x²) ＝ −12x³
Example ①
Calculate the following.
(1) 4x × (−3y) (2) 2a² × 5a (3) (−6xy) × (−2x)
Answer:
(1) −12xy、(2) 10a³、(3) 12x²y
Calculate the following.
(1) 4x × (−3y) (2) 2a² × 5a (3) (−6xy) × (−2x)
Answer:
(1) −12xy、(2) 10a³、(3) 12x²y
Division of Monomials
Write the expression as a fraction and cancel the same variables.
Example: 6x ÷ 2 ＝ 3x、12a²b ÷ 4ab ＝ 3a、(−15xy) ÷ 5x ＝ −3y
Example ②
Calculate the following.
(1) 8x²y ÷ 2xy (2) (−12a²b) ÷ (−4a)
Answer:(1) 4x、(2) 3ab
Calculate the following.
(1) 8x²y ÷ 2xy (2) (−12a²b) ÷ (−4a)
Answer:(1) 4x、(2) 3ab
Combined Multiplication and Division
When multiplication and division are mixed in an expression, change division into a fraction, or multiplication by the reciprocal, before calculating.
Example ③
Calculate the following: 6x²y × 4y ÷ 8xy
Answer:(6x²y × 4y) / (8xy) ＝ 24x²y² / 8xy ＝ 3xy。
Calculate the following: 6x²y × 4y ÷ 8xy
Answer:(6x²y × 4y) / (8xy) ＝ 24x²y² / 8xy ＝ 3xy。
Key Points
- Monomial × monomial: multiply the coefficients and variables separately
- Monomial ÷ monomial: write it as a fraction and cancel
- Mixed multiplication and division can be calculated at once as a fraction
- Check all signs together at the end
Practice Problems
- (−5a) × 3a²
- 18x²y ÷ (−6x)
- 4ab × 6a ÷ 3b
Answers and Explanations
- Answer:−15a³
Explanation: (−5)×3 = −15、a×a² = a³。 - Answer:−3xy
Explanation: 18x²y / (−6x) = −3xy。 - Answer:8a²
Explanation: 24a²b / 3b = 8a²。